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Peter Oberly, University of Rochester

"Some Bounds on the Arakelov-Zhang Pairing"

The Arakelov-Zhang pairing (also called the dynamical height pairing) is a kind of dynamical distance between two rational maps defined over a number field. This pairing has applications in arithmetic dynamics, especially as a tool to study the preperiodic points common to two rational maps. We will discuss some bounds on the Arakelov-Zhang pairing of f and g in terms of the coefficients of f and investigate some simple consequences of this result.   

MC 5417

Kieran Mastel, Department of Pure Mathematics, University of Waterloo

"An Aperiodic Monotile"

Last year, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss found the first example of an aperiodic monotile (or ‘einstein’), solving a longstanding open problem. We will look at the ‘hat’ tile they define and try to visually understand why it tiles the plane and why none of its tilings are periodic.

MC 5501

Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory VII"

We will discuss degree spectra of structures, following Antonio Montalbán's monograph.

MC 5479

Peter Pivovarov, University of Missouri

"A probabilistic approach to Lp affine isoperimetric inequalities"

In the class of convex sets, the isoperimetric inequality can be derived from several different affine inequalities. One example is the Blaschke-Santalo inequality on the product of volumes of a convex body and its polar dual. Another example is the Busemann--Petty inequality for centroid bodies. In the 1990s, Lutwak and Zhang introduced a related functional analytic framework with their notion of Lp centroid bodies, for p>1. Lutwak raised the question of encompassing the non-convex star-shaped range when p<1 (including negative values). I will discuss a probabilistic approach to establishing isoperimetric inequalities in this range. It uses a new representation of star-shaped sets as special averages of convex sets. Based on joint work with R. Adamczak, G. Paouris, and P. Simanjuntak.

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Andy Royston, Penn State University

"Solitons and the Extended Bogomolny Equations with Jumping Data"

The extended Bogomolny equations are a system of PDE's for a connection and a triplet of Higgs fields on a three-dimensional space. They are a hybrid of the Bogomolny equations and the Nahm equations. After reviewing how these latter systems arise in the study of magnetic monopoles, I will present an energy functional for which solutions of the extended Bogomolny equations are minimizers in a fixed topological class. In this construction, the connection and Higgs triplet are defined on all of R^3 and couple to additional dynamical fields localized on a two-plane that are analogous to jumping data in the Nahm equations. Solutions can therefore be interpreted as finite-energy BPS solitons in a three-dimensional theory with a planar defect. This talk is based on work done in collaboration with Sophia Domokos.

MC 5417

Anne Johnson, Department of Pure Mathematics, University of Waterloo

"Functor of Points and Examples"

We define and describe the functor of points. We then give examples of reduced schemes over algebraically closed fields from section 2.1 of Eisenbud and Harris.

MC 5417

Alina Stancu, Concordia University

"On the fundamental gap of convex domains in hyperbolic space"

The difference between the first two eigenvalues of the Dirichlet Laplacian on convex domains of R^n and, respectively S^n, satisfies the same strictly positive lower bound depending on the diameter of the domain. In work with collaborators, we have found that the gap of the hyperbolic space on convex sets behaves strikingly different even if a stronger notion of convexity is employed. This is very interesting as many other features of first two eigenvalues behave in the same way on all three spaces of constant sectional curvature. 

MC 5501

Xiao Zhong, Department of Pure Mathematics, University of Waterloo

"Green's Functions on the Berkovich Projective Line"

We introduce the green's functions and explore their properties. After this, we are ready to introduce a Bilu-type equidistribution theory in the next talk which is one of the main motivation for going deeply into this subject. The materials in this presentation cover the later half of the chapter 7 in Baker-Rumely's monograph on "Potential Theory and Dynamics on the Berkovich Projective Line".

MC 5417

Amanda Petcu, Department of Pure Mathematics, University of Waterloo

"An Introduction to (Lagrangian) Mean Curvature Flow"

In this talk, we will introduce the Mean Curvature Flow and explore some initial examples of the flow. We will show that in the compact case, the flow always produces singularities. We will also introduce type I and type II singularities. Finally, if time permits, we will discuss the Lagrangian Mean Curvature Flow and demonstrate that a mean curvature flow starting from a Lagrangian remains Lagrangian.

MC 5403

Guillermo Gallego, Universidad Complutense de Madrid

"Multiplicative Higgs bundles, monopoles and involutions"

Multiplicative Higgs bundles are a natural analogue of Higgs bundles on Riemann surfaces, where the Higgs field now takes values on the adjoint group bundle, instead of the adjoint Lie algebra bundle. In the work of Charbonneau and Hurtubise, they have been related to singular monopoles over the product of a circle with the Riemann surface.

In this talk we study the natural action of an involution of the group on the moduli space of multiplicative Higgs bundles, also from the point of view of monopoles. This provides a "multiplicative analogue" of the theory of Higgs bundles for real groups.

MC 5403